1 PRODUCTION SYSTEMS ENGINEERING Chapter 3: Mathematical Modeling of Production Systems Instructors: J. Li, S.M. Meerkov, and L. Zhang Copyright © 2008-2012 J. Li, S.M. Meerkov, and L Zhang

2 Motivation All methods of PSE are model-based. No two production systems are identical. It is possible, however, to reduce “any” production systems to one of the small set of standard models. These standard models and the process of reduction are the topics of this chapter. 3-2

3 OUTLINE 1. Types of Production Systems 2. Structural Modeling 3. Parametric Modeling: Mathematical Models of Machines 4. Parametric Modeling: Mathematical Models of Buffers 5. Modeling Interactions between Machines and Buffers 6. Performance Measures 7. Model Validation 8. Steps of Modeling, Analysis, Continuous Improvement, and Design 9. Simplification: Transforming Exponential Models into Bernoulli Ones 10. Case Studies 12. Summary 3-3

4 1. Types of Production Systems 1.1 Serial Lines 1.1.1 Basic structure Consists of processing units and MHD arranged serially: Processing unit: machine, manufacturing cell, shop in a plant, or a plant Material handling device: boxes, kanban cards, conveyors, AGVs, trucks, trains… The main feature: storing capacity. 3-4

5 1.1.2 Serial lines with FGB and carriers Serial lines with FGB: Closed serial lines: 3-5

6 1.1.3 Serial lines with quality issues Serial lines with inspection machines: Serial lines with rework: 3-6

7 1.1.4 Re-entrant lines (for semiconductor manufacturing) Serial line is a work horse of manufacturing: every production system contains a serial line. Large part of PSE is devoted to these systems. 3-7

8 1.2 Assembly Systems 1.2.1 Basic structure Two or more serial lines (component line) and one or more merge operations plus additional processing machines. 3-8

9 1.2.2 Assembly systems with multiple component lines and merge operations All variations in structure are possible for assembly systems as well. 3-9

10 2. Structural Modeling Structural modeling - the process of reduction of a production system to one of the standard types. Example: Serial lines with parallel or synchronous dependent machine. 3-10

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12 3-12

13 3-13

14 3-14

15 3. Parametric Modeling: Mathematical Models of Machines 3.1 Timing Issues 3.1.1 Machine cycle time and capacity Cycle time ( τ ) - time necessary to process a part by a machine. τ may be constant, variable or random. In large volume manufacturing it is mostly constant. Machine capacity ( c ) - number of parts produced per unit of time. 3-15

16 3.1.2 Slotted vs. unslotted time Slotted time : time is slotted with slot duration τ ; all transitions occur at the beginning or the end of a slot. Systems satisfying this property are called synchronous. Unslotted or continuous time : the above mentioned changes may occur at any time moment. If the cycle times of all machines are identical, such a system is still referred to as synchronous. If the cycle times are not identical, the system is called asynchronous. 3-16

17 3.1.3 Discrete event vs. flow systems Discrete event systems : only a complete job is transferred into or taken from a buffer. Flow systems : infinitesimal part of a job is transferred to or taken from a buffer. In the unslotted time case, flow systems are easier for analysis. 3-17

18 3.2 Machine Reliability Models Machine reliability model - pmf or pdf of up- and downtime 3.2.1 Reliability models for slotted time case: Bernoulli (B) Applicable to operations where downtime is comparable with cycle time. 3-18

19 3.2.1 Reliability models for slotted time case (cont.) Geometric (Geo) Machine states: Transition probabilities: 3-19

20 3.2.1 Reliability models for slotted time case (cont.) Resulting pmf’s of up- and downtime: Memoryless pmf. Applicable when the downtime is much longer than the cycle time (e.g., some machining operations) 3-20

21 3.2.1 Reliability models for slotted time case (cont.) Moments of slotted time reliability models 3-21

22 3.2.2 Reliability models for continuous time case Machines with a constant breakdown and repair rates 3-22

23 3.2.2 Reliability models for continuous time case (cont.) Calculation of uptime pdf’s: Assume. Then, by the total probability formula 3-23

24 3.2.2 Reliability models for continuous time case (cont) For δt → 0, 3-24

25 3.2.2 Reliability models for continuous time case (cont) The probability that the transition from up to down occurs during ( t, t + δt ): Thus, a constant breakdown rate results in exponential pdf of the uptime: A similar pdf (defined by the repair rate µ ) is obtained for the downtime. 3-25

26 3.2.2 Reliability models for continuous time case (cont) Applicable to machining operations with a very small “most- probable” uptime and downtimes. 3-26

27 3.2.2 Reliability models for continuous time case (cont) Machines with linearly increasing breakdown and repair rates. Using the same approach, we derive which is the Rayleigh pdf. 3-27

28 3.2.2 Reliability models for continuous time case (cont) Applicable to machining operations where “most-probable” up- and downtimes are away from 0. 3-28

29 3.2.2 Reliability models for continuous time case (cont) Machines with decreasing breakdown and repair rates long tail 3-29

30 3.2.2 Reliability models for continuous time case (cont) Weibull reliability model 3-30

31 3.2.2 Reliability models for continuous time case (cont) Gamma reliability model 3-31

32 3.2.2 Reliability models for continuous time case (cont) Log-normal reliability model Consider, where X is a Gaussian random variable. Then, t up has the log-normal pdf: 3-32

33 3.2.2 Reliability models for continuous time case (cont) Mixed reliability model (M): up- and downtime are distributed differently with pdf’s from the set {exp, Ra, W, ga, LN}. General reliability model (G): up- and downtime have arbitrary distribution. 3-33

34 3.2.2 Reliability models for continuous time case (cont) Moments of continuous time reliability models 3-34

35 3.2.3 Efficiency of the machines For any reliability model, machine efficiency ( e ) is the probability to produce a part during a cycle time. Efficiency of machines with various reliability models are as follows: Bernoulli: e = p 3-35

36 3.2.3 Efficiency of the machines (cont) Geometric Markov balance equations: Steady state probabilities: Since 3-36

37 3.2.3 Efficiency of the machines (cont) Exponential Markov balance equations: Steady state probabilities: Since 3-37

38 3.2.3 Efficiency of the machines (cont) General Let t up and t down be random up- and downtime distributed arbitrarily with expected values T up and T down. Assume that t up and t down are measured in units of cycle time. Let Then 3-38

39 3.2.3 Efficiency of the machines (cont) According to the strong law of large numbers, the following takes place with probability 1: Thus, w p.1 T up e = T +T+T down 3-39

40 3.2.4 Coefficients of variations for up- and downtime of manufacturing equipment Manufacturing equipment on the factory floor often has coefficients of variation of up- and downtime less than 1. The explanation of this phenomenon is as follows: Theorem: If the breakdown (respectively, repair) rate is increasing in time, the resulting uptime (respectively, downtime) distribution has the coefficient of variation less than 1. If the breakdown (respectively, repair) rate is decreasing in time, the coefficient of variation is greater than 1. 3-40

41 3.2.5 Time dependent vs. operation-dependent failures Time-dependent failures - machine breakdown can occur even the machine is blocked or starved. Operation-dependent failures - machine breakdown occur only when machine is not blocked or starved. Both are practical. The difference in performance is small. In this course, we use time-dependent failures (easier to analyze). 3-41

42 3.3 Notations Machines: belong to { exp, Ra, W, ga, LN, G } Synchronous serial line: Asynchronous serial line: 3-42

43 3.4 Machine Model Identification Goal: Determine τ : use “stop watch” (include loading/unloading time) andare practically impossible to identify. 3-43

44 3.4.1 What is possible (and necessary!) to identify 3-44

45 3.4.2 Aggregating parallel machines for structural modeling Identical machines case: m i : { τ, T up, T down }, i = 1,…, S. 3-45

46 3.4.2 Aggregating parallel machines for structural modeling (cont) Non-identical machines case: m i : { τ i, T up,i, T down,i }, i = 1,…, S. 3-46

47 3.4.2 Aggregating parallel machines for structural modeling (cont) Example Original system 3-47

48 3.4.2 Aggregating parallel machines for structural modeling (cont) Aggregated system Accuracy 3-48

49 3.4.3 Aggregating consecutive dependent machines for structural modeling Identical machines case: m i : { τ, T up, T down }, i = 1,…, S. 3-49

50 3.4.3 Aggregating consecutive dependent machines for structural modeling(cont.) Non-identical machines case: m i : { τ, T up, T down }, i = 1,…, S. 3-50

51 3.4.4 PSE Toolbox 3-51

52 3.4.4 PSE Toolbox (cont.) Modeling function and tools 3-52

53 3.4.4 PSE Toolbox (cont.) Aggregation of parallel machines 3-53

54 3.4.4 PSE Toolbox (cont.) Aggregation of consecutive dependent machines 3-54

55 3.5 Machine Quality Models Machine quality model - pmf or pdf of time intervals when defectives are produced. Bernoulli quality model Exponential model General g 3-55

56 4. Parametric Modeling: Mathematical Models of Buffers 4.1 Structural Model MHD identification: identification of { τ, T up, T down } of the machines part and identification of N. Often, mechanical part may be omitted. 3-56

57 4.2 Modeling Conveyors Number of parts to sustain continuous operation Buffer capacity K i - number of parts that fit into the conveyor 3-57

58 5. Modeling Interaction between Machines and Buffers 5.1 Slotted Time Case 5.1.1 State changing conventions Machine state is determined at the beginning of each time slot. Buffer state is determined at the end of each time slot. 3-58

59 5.1.2 Blocking and starvation conventions Blocked before service (BBS) - a machine cannot operate if it is up, the downstream buffer is full and the downstream machine does not take a part from this buffer → The part in the machine is viewed as if it is in the buffer. Blocked after service (BAS) - if up and not starved, the machine operates even if the downstream buffer is full. This implies that Starvation - a machine is starved if it is up and the upstream buffer is empty. 3-59

60 6. Performance Measures 6.1 Production Rate and Throughput Production rate ( PR ) - average number of parts produced by the last machine per cycle time in the steady state. Appropriate for synchronous systems. Throughput ( TP ) - average number of parts produced by the last machine per unit of time in the steady state. Appropriate for both synchronous and asynchronous systems. 3-60

61 6.1 Production Rate and Throughput (cont)  Functional dependence of PR : For a Bernoulli line: For a synchronous exponential line: For an asynchronous exponential line: For a synchronous line with Weibull machines: For a closed Bernoulli line: 3-61

62 6.1 Production Rate and Throughput (cont) For a Bernoulli assembly system: For a line with Bernoulli quality and reliability model: None of these function can be evaluated in closed form for M > 2. For Bernoulli and exponential reliability models, these function can be evaluated analytically using recursive procedures. For other cases, empirical formulas can be derived. 3-62

63 6.2 Work-in-process and Finished Goods Inventories WIP i - average number of parts, i.e., i -th buffer, in the steady state. Finished goods inventory ( FGI ) - average number of parts in FGB in steady state. For Bernoulli lines: For exponential lines: 3-63

64 6.3 Probabilities of Blockages and Starvations Blockage of m i ( BL i ) - steady state probability that m i is up, b i is full and m i +1 does not take a part from b i. Starvation of m i ( ST i )- steady state probability that m i is up and b i -1 is empty. For slotted time case: 3-64

65 6.3 Probabilities of Blockages and Starvations (cont) For continuous time case: For exponential lines: Closed form expressions are available only for Markovian systems with M = 2. For M > 2, recursive procedures are derived 3-65

66 6.4 Due-time Performance DTP - the steady state probability to ship to the customer the desired number of parts, D, during a given shipping period, T : For exponential line Analytical expression is available only for M = 1. For M > 1, lower bound is available. 3-66

67 6.5 Transient Characteristics Transient characteristics - a group of performance measures that describe how PR and WIP reach their steady states: t sPR - settling time for PR t sWIP - settling time for WIP All are characterized by the second largest eigenvalues of the transition matrix and pre-exponential coefficients. Solution are available for Markovian cases. 3-67

68 6.6 Residence Time 3-68

69 6.7 Measuring Performance Characteristics of the Factory Floor PR and TP are typically monitored. WIP i and RT less often. BL i and ST i very seldom - although, as we will see later, they are most important for system “diagnostics” and continuous improvement. 3-69

70 7. Model Validation Model validation - the process of assessing accuracy of the mathematical model. Measures of accuracy: Acceptable values. Since the parameters of the machines and buffers are typically known with low accuracy (±5%), ε PR within ±5% is typically acceptable. 3-70

71 8. Steps of Modeling, Analysis, Continuous Improvement, and Design 8.1 Modeling Layout investigation Structural modeling Parametric modeling Machine parameter identification Buffer parameter identification Model validation 3-71

72 8.2 Analysis, Continuous Improvement, and Design Using the mathematical model of a production system, the following problem can be solved: Evaluation of PR, TP, WIP, BL, ST, etc. Investigation of “what if” scenarios Direction for improvement and design of continuous improvement projects with predictable results 3-72

73 9. Simplification: Transforming Exponential Models into Bernoulli Ones 9.1 Scenario Motivation: Bernoulli models are easier to deal with. exp-B transformation B-exp transformation 3-73

74 9.2 Exp-B Transformation Expressions where 3-74

75 Accuracy M = 3, λ = 0.01, μ = 0.1 3-75

76 9.3 B-exp Transformation Expressions Accuracy - the same as exp-B. 3-76

77 9.3 PSE Toolbox Exp-B transformation 3-77

78 9.3 PSE Toolbox Exp-B-Exp transformation: 3-78

79 10. Case Studies 10.1 Automotive Ignition Coil Processing System 10.1.1 System layout 3-79

80 10.1.2 System performance 3-80

81 10.1.3 Structural modeling 3-81

82 10.1.4 Modeling and identification of the machines Exponential assumption 3-82

83 10.1.5 Modeling and identification of the buffers 3-83

84 10.1.6 Overall system model Exponential 3-84

85 10.1.6 Overall system model (cont) Bernoulli 3-85

86 10.2 Automotive Paint Shop Production System 10.2.1 System layout 3-86

87 10.2.2 System performance 3-87

88 10.2.3 Structural modeling 3-88

89 10.2.4 Modeling and identification of the machines Average production losses (jobs/hour) 3-89

90 10.2.4 Modeling and identification of the machines (cont) Bernoulli assumption 3-90

91 10.2.5 Modeling and identification of the buffers 3-91

92 10.2.6 Overall system model 3-92

93 10.3 Automotive Ignition Module Assembly System 10.3.1 System layout 3-93

94 10.3.2 System performance Numerical throughput 600 part/hour. Losses ≈ 40%. 3-94

95 10.3.3 Structural modeling 3-95

96 10.3.3 Structural modeling (cont.) Exponential assumption τ i ≈ 6 sec/part 3-96

97 10.3.4 Modeling and identification of the buffers 3-97

98 10.3.5 Average frequency of starvations and blockages by carriers 3-98

99 10.3.6 Overall system model Exponential 3-99

100 10.3.6 Overall system model (cont) Bernoulli 3-100

101 11. Summary 11.1 Steps of Mathematical Modeling Layout Structural modeling Parametric modeling Machine identification Buffer identification Model validation 3-101

102 11.2 Machine Model Cycle time pmf or pdf of uptime pmf or pdf of downtime pmf or pdf of parts quality 3-102

103 11.3 Reliability Models Bernoulli Geometric Exponential Rayleigh Weibull Gamma Log-normal General 3-103

104 11.4 Modeling Conventions Slotted or unslotted time Time-dependent or operation-dependent Synchronous or asynchronous Discrete event or flow model Blocked before or blocked after service 3-104

105 11.5 Performance Measures Production rate or throughput Work-in-process Finished goods inventory Probability of blockages and starvations Customer demand satisfaction Transient properties 3-105